PAMM · Proc. Appl. Math. Mech. 7, 3030003–3030004 (2007) / DOI 10.1002/pamm.200700358
Modeling, simulation and control of a doubly-fed induction machine
Carles Batlle∗1,2 , Arnau Dòria-Cerezo∗∗1,3 , and Enric Fossas∗∗∗1
1
2
3
Institute of Industrial and Control Engineering, Universitat Politècnica de Catalunya, Spain.
Dept. of Applied Mathematics IV, EPSEVG, Universitat Politècnica de Catalunya, Vilanova i la Geltrú, Spain.
Dept. of Electrical Engineering, EPSEVG, Universitat Politècnica de Catalunya, Vilanova i la Geltrú, Spain.
The Hamiltonian model and control of a doubly-fed induction machine is discussed. The geometric properties of the Hamiltonian formalism allow us to develop control algorithms which assure stability and good performance. Experimental results
in good accordance with the control goals are presented.
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 The doubly-fed induction machine
Doubly-fed induction machines (DFIMs) have been proposed in the literature, among other applications, for high-performance
storage systems, wind turbine generators or hybrid engines (see [1] for and extended literature survey and discussion). The
attractiveness of the DFIM stems primarily from its ability to handle large-speed variations around the synchronous speed.
Furthermore, this induction machine require an small power electronic equipment for control purposes (instead of the classical
induction machines). The DFIM is a three-phase induction machine whit both stator and rotor windings accessible, and the
rotor side is used for control.
A DFIM can be used as a generator or motor. In this work we present a controller for a motor mode, and the goals are to
regulate the mechanical speed while keeping the power factor close to one.
Under the assumption that the machine is symmetric (all windings are equal), the cross inductances are smooth, sinusoidal functions of θ (rotor position), and the three-phase system is balanced, this mathematical transformation can be used
to decouple variables (reduce the system dimension) and to facilitate the solution of equations with time-varying coefficients.
Then, the original θ-depending model is simplified, and also, for control purposes, the dq, or Park, transformation [2] allows to describe a tracking problem as a regulation one, which can be solved with IDA-PBC (Interconnection and Damping
Assignment–Passivity-based Control) techniques [3].
2 Port-Hamiltonian model of a DFIM
As discussed in [4] (and references therein) a large class of physical systems of interest in control applications can be modelled
in the general form of Port-controlled Hamiltonian systems (PCHS). An explicit PCHS has the form
ẋ = (J(x) − R(x))∂x H(x) + g(x)u
(1)
y = g T (x)(∂x H(x))T
where x ∈ Rn is the vector state, or Hamiltonian variables, y ∈ Rm are the port variables, H(x) : Rn → R is the Hamiltonian
function (or energy function), J(x) ∈ Rn×n is the interconnection matrix (J(x) = −J(x)T ), R(x) ∈ Rn×n is the dissipation
matrix (R(x) = RT ≥ 0) and g(x) ∈ Rn×m is the external port connection matrix.
The dq model of a DFIM can be written as a PCHS (1), with the Hamiltonian variables xT = (λTs , λTr , Jm ω) ∈ R5 , where
λs , λr are the stator and rotor fluxes, ω is the mechanical speed, Jm is the inertia of the rotating parts, and
⎡
⎤
⎤
⎡
−ωs Ls J2
−ωs Lsr J2
O2×1
Rs I2 O2×2 O2×1
0 −1
⎣
⎦
⎦
⎣
J(x) = −ωs Lsr J2 −(ωs − ω)Lr J2 Lsr J2 is , R(x) = O2×2 Rr I2 O2×1 , J2 =
, (2)
1 0
O1×2
Lsr iTs J2
0
O1×2 O1×2
Br
where L are inductances and R are resistances (lower indices s and r refer to stator and rotor, respectively), Br is the mechanical damping, is , ir ∈ R2 are the stator and rotor currents and I2 a 2×2 identity matrix. The port variables are uT = [vsT , vrT , τL ]
and the interconnection matrix g is a 5 × 5 identity matrix. Furthermore
1
1 T T
λs
is
Ls I2 Lsr I2
is
λs
(3)
+ Jm ω 2 .
=L
=
, H = [λs , λr ]L
λ
λr
ir
Lsr I2 Lr I2
ir
2
2
r
∗
e-mail: carles.batlle@upc.edu.
Corresponding author: e-mail: arnau.doria@upc.edu, Phone: +34 93 401 25 97, Fax: +34 93 401 66 05.
∗∗∗ e-mail: enric.fossas@upc.edu.
∗∗
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM Minisymposia
3030004
3 Port-Hamiltonian control of a DFIM
Most DFIM controllers proposed in the literature are based on vector control and decoupling [5]. Vector control contains an
inner-loop which requires the reconstruction of the magnetic flux variables (which are not directly measurable).
Passivity-based Control (PBC) [6] uses the fact that passive nonlinear systems are described by an storage function (which
is a proper Lyapunov function). The control design main goal is then to reshape the original energy function by means of
the controller. Based on PBC, the IDA-PBC technique, which uses the passive properties of port-Hamiltonian Systems, was
presented in [3].
In [7] it is shown that the standard procedure using IDA-PBC can be applied for controlling the DFIM, treating the mechanical system as a cascaded subsystem. Using the SIDA-PBC (Simultaneous IDA-PBC) approach [8], it is possible to shape
the energy function of the complete system, resulting in a controller with improved power flow regulation performance. Further improvements of the basic IDA-PBC [1] allow to design a controller with reduced parameter dependence [9]. The new
controller is more robust than the previous ones based on basic (S)IDA-PBC. The main advantages with respect to classical
vector control are:
• The stator flux is not required. This dispenses with the estimation of a critical variable which will be used to construct
the rotating reference.
• The rotating reference is referred to the stator voltage, which, as discussed in the previous item, is easier to obtain than
the stator flux oriented reference. The computations depend only on a measured variables.
• The reference inputs of the new controller are directly the stator currents (which are the outputs of the electrical subsystem), whereas the references of the inner-loop of vector control are rotor currents.
• No assumption is made on the smallness of rotor resistances.
Experimental results [9] are presented in Fig. 1. In this case the mechanical speed (Fig. 1a) is regulated with a good power
factor (Fig. 1b), i.e. voltage and current are nearly in phase.
Stator voltage and current, a component
Mechanical speed
330
300
300
325
200
200
Vsa[V], isa[A]*200
w[rad/s]
100
100
320
315
0
0
−100
−100
−200
−200
310
305
300
951
−300
957.3
952
953
954
955
956
957
time[s]
958
959
960
961
962
a)
−300
957.32
957.34
957.36
957.38
957.4
time[s]
957.42
957.44
957.46
957.48
957.32
957.324
957.328
time[s]
957.332
957.336
b)
Fig. 1 Experimental results. a) Mechanical speed, ω, b) stator voltage an current (Vsa , isa ).
Acknowledgements The authors have been partially supported by the Spanish project DPI2004-06871-CO2-02.
References
[1] A. Dòria-Cerezo. Modeling, simulation and control of a doubly-fed induction machine controlled by a back-to-back converter (PhD.
Thesis, Universitat Politècnica de Catalunya, 2006).
[2] P.C. Krause, O. Wasynczuk, and S.D. Sudhoff. Analysis of Electric Machinery and Drive Systems (John Wiley & Sons Inc., 2002).
[3] R. Ortega, A. van der Schaft, B. Maschke and G. Escobar. Interconnection and damping assignment passivity-based control of portcontrolled Hamiltonian systems. Automatica 38(4), 585-596 (2002).
[4] A. van der Schaft. L2 gain and passivity techniques in nonlinear control (Springer, 2000).
[5] W. Leonard. Control of electric drives (Springer, 1995).
[6] R. Ortega, A. Loria, P.J. Nicklasson and H. Sira-Ramirez. Passivity-based control of Euler-Lagrange systems (Springer-Verlag, 1998).
[7] C. Batlle, A. Dòria-Cerezo, and R. Ortega. Power Flow Control of a Doubly–Fed Induction Machine Coupled to a Flywheel. European
Journal of Control 11(3), 209-221 (2005).
[8] C. Batlle, A. Dòria-Cerezo, G. Espinosa, and R. Ortega. Simultaneous Interconnection and Damping Assignment Passivity-Based
Control: Two Practical Examples, in: 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, (2006).
[9] C. Batlle, A. Dòria-Cerezo, and R. Ortega. A Robustly Stable PI Controller for the Doubly-fed Induction Machine, in: Proc. 32nd
Annual Conference of the IEEE Industrial Electronics Society (IECON), (2006), pp. 5113-5118.
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim